To solve for y in the equation 2y² - 2y + 4 = (y + 1)², follow these steps:
Step 1: Start by expanding the expression (y + 1)². This gives y² + 2y + 1.
Step 2: Rewrite the equation substituting (y + 1)² with its expanded version. The updated equation is 2y² - 2y + 4 = y² + 2y + 1.
Step 3: To simplify the equation, combine like terms. Take all terms involving y to one side of the equation and the constants to the other. The updated equation is y² - 4y + 3 = 0.
Step 4: The equation y² - 4y + 3 = 0 is a quadratic equation. Apply the quadratic formula to find the solutions for y. The quadratic formula is given by y = [-b ± sqrt(b² - 4ac)] / (2a), where a, b, and c are the coefficients of y², y, and the constant term respectively. For this equation, a = 1, b = -4, and c = 3.
Step 5: Using these values in the quadratic formula, we find that y = [4 ± sqrt((-4)² - 4*1*3)] / 2*1.
Solving this gives two possible solutions for y, which are y = 1 and y = 3.
So, the solutions for the equation 2y² - 2y + 4 = (y + 1)² are y = 1, 3.